ORBIFOLDS OF SYMPLECTIC FERMION ALGEBRAS

THOMAS CREUTZIG AND ANDREW R. LINSHAW

ABSTRACT. We present a systematic study of the orbifolds of the rank n symplectic fermionalgebra A(n), which has full automorphism group Sp(2n). First, we show that A(n)Sp(2n)

and A(n)GL(n) are W-algebras of type W(2, 4, . . . , 2n) and W(2, 3, . . . , 2n+1), respectively.Using these results, we find minimal strong finite generating sets for A(mn)Sp(2n) andA(mn)GL(n) for all m,n ≥ 1. We compute the characters of the irreducible representationsof A(mn)Sp(2n)×SO(m) and A(mn)GL(n)×GL(m) appearing inside A(mn), and we expressthese characters using partial theta functions. Finally, we give a complete solution to theHilbert problem for A(n); we show that for any reductive group G of automorphisms, A(n)G

is strongly finitely generated.

1. INTRODUCTION

Vertex algebras are a fundamental class of algebraic structures that arose out of con-formal field theory and have applications in a diverse range of subjects. Given a vertexalgebra V and a group G of automorphisms of V , the invariant subalgebra VG is called anorbifold of V . Many interesting vertex algebras can be constructed either as orbifolds or asextensions of orbifolds. A spectacular example is the Moonshine vertex algebra V , whosefull automorphism group is the Monster, and whose graded character is j(τ)− 744 wherej(τ) is the modular invariant j-function [FLM].

In physics, rational vertex algebras, of which V is an example, correspond to rationaltwo-dimensional conformal field theories. Other well-known examples include the Vi-rasoro minimal models [GKO, WaI], affine vertex algebras at positive integer level [FZ],lattice vertex algebras associated to positive-definite even lattices [D], and certain familiesof W-algebras [ArI, ArII]. A rational vertex algebra V has only finitely many irreducible,admissible modules, and any admissible V-module is completely reducible. On the otherhand, many interesting vertex algebras admit modules which are reducible but indecom-posable. Following [CR], we shall call such vertex algebras logarithmic, as the correspond-ing conformal field theories often have logarithmic singularities in their correlation func-tions. For many years after Zhu’s thesis appeared in 1990 [Zh], it was believed that ratio-nality and the C2-cofiniteness conditions were equivalent; see for example [DLMII, ABD].However, this was disproven by the construction of the Wp,q-triplet algebras [KI, FGST],which are logarithmic and C2-cofinite [A, AM, TW].

One of the first logarithmic conformal field theories studied in the physics literaturewas the symplectic fermion theory [KII]. The symplectic fermion algebra A(n) of rank nhas odd generators ei, f i for i = 1, . . . , n satisfying operator product expansions

ei(z)f j(w) ∼ δi,j(z − w)−2.

Key words and phrases. free field algebra, symplectic fermions, invariant theory; reductive group action;orbifold; strong finite generation; W-algebra; character formula; theta function.

1

It is an important example of a free field algebra, and the simplest triplet algebra W2,1

coincides with the orbifold A(1)Z/2Z. It is also known that A(n)Z/2Z is logarithmic andC2-cofinite [A], but little is known about the structure and representation theory of moregeneral orbifolds of A(n).

Our goal in this paper is to carry out a systematic study of the orbifolds A(n)G, where Gis any reductive group of automorphisms of A(n). The full automorphism group of A(n)is the symplectic group Sp(2n). In order to describe A(n)G for a general G, a detailedunderstanding of the structure and representation theory of A(n)Sp(2n) is necessary. Aswe shall see, all irreducible, admissible A(n)Sp(2n)-modules are highest-weight modules,and A(n)G is completely reducible as an A(n)Sp(2n)-module.

Our study of A(n)Sp(2n) is based on classical invariant theory, and follows the approachdeveloped in [LI, LII, LIII, LIV, LV]. First, A(n) admits an Sp(2n)-invariant filtration suchthat gr(A(n)) ∼=

∧⊕k≥0 Uk as supercommutative rings. Here each Uk is a copy of the

standard Sp(2n)-representation C2n. As a vector space, A(n)Sp(2n) is isomorphic to R =(∧⊕k≥0 Uk

)Sp(2n), and we have isomorphisms of graded commutative rings

gr(A(n)Sp(2n)) ∼= gr(A(n))Sp(2n) ∼= R.

In this sense, we regard A(n)Sp(2n) as a deformation of the classical invariant ring R.By an odd analogue of Weyl’s first and second fundamental theorems of invariant the-

ory for the standard representation of Sp(2n), R is generated by quadratics

qa,b| 0 ≤ a ≤ b,and the ideal of relations among the qa,b’s is generated by elements

pI | I = (i0, . . . , i2n+1), 0 ≤ i0 ≤ · · · ≤ i2n+1of degree n+ 1, which are analogues of Pfaffians. We obtain a corresponding strong gen-erating set ωa,b for A(n)Sp(2n), as well as generators PI for the ideal of relations amongthe ωa,b’s, which correspond to the classical relations with suitable quantum corrections.In fact, there is a more economical strong generating set j2k = ω0,2k| k ≥ 0, and the setsωa,b| 0 ≤ a ≤ b and ∂ij2k| i, k ≥ 0 are related by a linear change of variables. The re-lation of minimal weight among the generators occurs at weight 2n+ 2, and correspondsto I = (0, 0, . . . , 0).

A key technical result in this paper is the analysis of the quantum corrections of theabove classical relations. For each pI , there is a certain correction term RI appearing in PI

which we call the remainder. We show that RI satisfies a recursive formula which impliesthat the relation of minimal weight has the form

(1.1) j2n = Q(j0, j2, . . . , j2n−2).

Here Q is a normally ordered polynomial in j0, j2, . . . j2n−2 and their derivatives. We call(1.1) a decoupling relation, and by applying the operator j21 repeatedly, we can constructhigher decoupling relations

j2m = Qm(j0, j2, . . . , j2n−2)

for all m > n. This shows that j0, j2, . . . , j2n−2 is a minimal strong generating set forA(n)Sp(2n), and in particular A(n)Sp(2n) is of type W(2, 4, . . . , 2n); see Theorem 3.9. Us-ing this result, we give a minimal strong generating set for A(mn)Sp(2n) for all m,n ≥

2

1. Similarly, using the invariant theory of GL(n) we show that A(n)GL(n) is of typeW(2, 3, . . . , 2n + 1), and we give a minimal strong generating set for A(mn)GL(n) for allm,n ≥ 1.

Character decompositions. By a general theorem of Kac and Radul [KR], A(mn) decom-poses as a sum of irreducible A(mn)G-modules for every reductive group G ⊂ Sp(2mn)(see also [DLMI]). Such decompositions are very useful for computing the characters oforbifolds of free field algebras; see for example [WaII, KWY]. More recently, the char-acters of some orbifolds of the βγ-system of rank n were computed in [BCR]. The taskwas to find Fourier coefficients of certain negative index meromorphic Jacobi forms, andone way to solve the decomposition problem used the denominator identity of affine Liesuperalgebras. We will instead use the denominator identity of the finite-dimensional Liesuperalgebra spo(2n|m) to find the character decomposition when G = Sp(2n)× SO(m),and the denominator identity of gl(n|m) to find the character decomposition for G =GL(n) × GL(m). In all these cases we obtain explicit character formulas, and they areexpressed using partial theta functions. The character of A(mn) is

qmn12

∞∏j=1

(1 + qj)2mn =

(η (q2)

η (q)

)2mn

with the Dedekind eta function η(q) = q124

∞∏n=1

(1− qn).

In order to perform the character decomposition, we need to refine the grading. Forthis let g0 = gl(m) ⊕ gl(n) or g0 = sp(2m) ⊕ so(n) be the even subalgebra of g = gl(m|n),respectively g = spo(2m|n). Then the set of odd roots ∆1 of g is contained in the weightlattice of g0, and the A(mn)-algebra character, graded by both the weight lattice of g0 andby conformal dimension, is

ch[A(mn)] = qmn12

∏α∈∆1

∞∏n=1

(1 + eαqn) .

The character can be rewritten as

ch[A(mn)] = e−ρ1qmn12

∏α∈∆+

1

eα2

(1 + e−α)

∞∏n=1

(1 + eαqn)(1 + e−αqn−1

)=

1

eρ1

∏α∈∆+

1

ϑ (eα; q)

(1 + e−α) η(q),

where ρ1 is the odd Weyl vector of g, ∆+1 is the set of its positive odd roots and

ϑ(z; q) =∑n∈Z

zn+12 q

12(n+

12)

2

is a standard Jacobi theta function. Let P+ be the set of dominant weights of the evensubalgebra g0. Then our result, Theorem 5.4, states that the graded character of A(mn)decomposes as

ch[A(mn)] =∑

Λ∈L∩P+

chΛBΛ.

3

Here chΛ is the irreducible highest-weight representation of g0 of highest-weight Λ. Thebranching function is

BΛ =1

η(q)|∆+1 |

|W ♯||W |

∑ω∈W

ϵ(w)∑

((nα),(mβ))∈Iω(Λ+ρ0)−ρ0

q12(nα+

12)

2

Pmβ(q)

and

Pn(q) := q12(n+

12)

2∞∑

m=0

(−1)mq12(m2−2m(n+ 1

2))

is a partial theta function. Further, ρ0 is the Weyl vector of g0, L is the root lattice of g, Wis its Weyl group and W ♯ the subgroup of W corresponding to the larger subalgebra ofg0 and finally Iλ is a subset of Z|∆+

1 |. More details on these objects are found in Section5. Summing over all representations corresponding to either so(m) or gl(m), one thenobtains the character decomposition for G = Sp(2n), and respectively for G = GL(n).In particular, the character formula for A(n)Sp(2n) implies that it is freely generated; thereare no nontrivial normally ordered relations among the generators j0, j2, . . . , j2n−2. Thisyields an explicit classification of the irreducible, admissible A(n)Sp(2n)-modules.

One of the important properties of rational vertex algebras is that characters of modulesare the components of a vector-valued modular form for the modular group SL(2,Z)[Zh]. In the non-rational case, the relation to modularity is unclear. However, there areexamples of affine vertex superalgebras whose module characters are built out of mockmodular forms [KWI]. For the singlet vertex algebra W(2, 2p − 1) for p ≥ 2, the modulecharacters are, up to a prefactor of η(q)−1, partial theta functions [F], and W(2, 3) coincideswith A(1)GL(1). The characters are then not modular, but their modular group action isstill known [Zw, CM]. They carry an infinite-dimensional modular group representationthat is compatible with the expectations of vertex algebras and conformal field theory inthe cases of a family of vertex superalgebras [AC] and the singlet vertex algebras [CM].

The Hilbert problem for A(n). A vertex algebra V is called strongly finitely generated ifthere exists a finite set of generators such that the set of iterated Wick products of thegenerators and their derivatives spans V . Recall Hilbert’s theorem that if a reductivegroup G acts on a finite-dimensional complex vector space V , the ring O(V )G of invariantpolynomial functions is finitely generated [HI, HII]. The analogous question for vertexalgebras is the following. Given a simple, strongly finitely generated vertex algebra Vand a reductive group G of automorphisms of V , is VG strongly finitely generated? Thisproblem was solved affirmatively when V is the Heisenberg vertex algebra H(n) [LIII,LIV], and when V is the βγ-system S(n) or free fermion algebra F(n) [LV]. The approachis the same in all these cases and is based on the following observations.

(1) The Zhu algebra of VAut(V) is abelian, which implies that its irreducible, admissiblemodules are all highest-weight modules.

(2) For any reductive G, VG has a strong generating set that lies in the direct sum offinitely many irreducible VAut(V)-modules.

(3) The strong finite generation of VAut(V) implies that these modules all have a certainfiniteness property.

4

In the case V = A(n), these conditions are satisfied, so the same approach yields a solutionto the Hilbert problem for A(n); see Theorem 6.7. The method is essentially constructiveand it reveals how A(n)G arises as an extension of A(n)Sp(2n) by irreducible modules.

A free field algebra is any vertex algebra of the form

H(n)⊗F(m)⊗ S(r)⊗A(s), m, n, r, s ≥ 0,

where V(0) is declared to be C for V = H,S,F ,A. Many interesting vertex algebrashave free field realizations, that is, embeddings into such algebras. In addition, manyvertex algebras can be regarded as deformations of free field algebras. For example, ifg = g0 ⊕ g1 is a Lie superalgebra with a nondegenerate, supersymmetric bilinear form,the corresponding affine vertex superalgebra is a deformation of H(n) ⊗ A(m), wheren = dim(g0) and 2m = dim(g1). By combining Theorem 6.7 with the results of [LIII,LIV, LV], one can establish the strong finite generation of a broad class of orbifolds of freefield algebras, as well as their deformations. This has an application to the longstandingproblem of describing coset vertex algebras, which appears in a separate paper [CL].

2. VERTEX ALGEBRAS

We shall assume that the reader is familiar with the basics of vertex algebra theory,which has been discussed from various points of view in the literature (see for example[B, FLM, K, FBZ]). We will follow the formalism developed in [LZ] and partly in [LiI], andwe will use the notation of the previous paper [LI] of the second author. In particular, bya vertex algebra, we mean a quantum operator algebra A in which any two elements a, bare local, meaning that (z − w)N [a(z), b(w)] = 0 for some positive integer N . This is wellknown to be equivalent to the notion of vertex algebra in [FLM]. The operators productexpansion (OPE) formula is given by

a(z)b(w) ∼∑n≥0

a(w) n b(w) (z − w)−n−1.

Here ∼ means equal modulo terms which are regular at z = w, and n denotes the nth

circle product. A subset S = ai| i ∈ I of A is said to generate A if A is spanned by wordsin the letters ai, n, for i ∈ I and n ∈ Z. We say that S strongly generates A if every A isspanned by words in the letters ai, n for n < 0. Equivalently, A is spanned by

: ∂k1ai1 · · · ∂kmaim : | i1, . . . , im ∈ I, k1, . . . , km ≥ 0.

We say that S freely generates A if there are no nontrivial normally ordered polynomialrelations among the generators and their derivatives.

Our main example in this paper is the symplectic fermion algebra A(n) of rank n, whichis freely generated by odd elements ei, f i| i = 1, . . . , n satisfying

ei(z)f j(w) ∼ δi,j(z − w)−2, f j(z)ei(w) ∼ −δi,j(z − w)−2,

ei(z)ej(w) ∼ 0, f i(z)f j(w) ∼ 0.(2.1)

We give A(n) the conformal structure

(2.2) LA = −n∑

i=1

: eif i :

5

of central charge −2n, under which ei, f i are primary of weight one. The full automor-phism group of A(n) is the symplectic group Sp(2n). It acts linearly on the generators,which span a copy of the standard Sp(2n)-module C2n.

Category R. Let R be the category of vertex algebras A equipped with a Z≥0-filtration

(2.3) A(0) ⊂ A(1) ⊂ A(2) ⊂ · · · , A =∪k≥0

A(k)

such that A(0) = C, and for all a ∈ A(k), b ∈ A(l), we have

(2.4) a n b ∈

A(k+l) n < 0A(k+l−1) n ≥ 0

.

Elements a(z) ∈ A(d) \ A(d−1) are said to have degree d.Filtrations on vertex algebras satisfying (2.4) were introduced in [LiII], and are known

as good increasing filtrations. Setting A(−1) = 0, the associated graded object gr(A) =⊕k≥0 A(k)/A(k−1) is a Z≥0-graded associative, (super)commutative algebra, equipped with

a derivation ∂ of degree zero. For each r ≥ 1 we have the projection

(2.5) ϕr : A(r) → A(r)/A(r−1) ⊂ gr(A).

The assignment A 7→ gr(A) is a functor from R to the category of ∂-rings, i.e., Z≥0-graded(super)commutative rings with a differential ∂ of degree zero. A ∂-ring A is said to begenerated by a subset ai| i ∈ I if ∂kai| i ∈ I, k ≥ 0 generates A as a ring. The keyfeature of R is the following reconstruction property [LL].

Lemma 2.1. Let A be a vertex algebra in R and let ai| i ∈ I be a set of generators for gr(A) as a∂-ring, where ai is homogeneous of degree di. If ai(z) ∈ A(di) are elements such that ϕdi(ai(z)) =ai, then A is strongly generated as a vertex algebra by ai(z)| i ∈ I.

There is a similar reconstruction property for kernels of surjective morphisms [LI]. Letf : A → B be a morphism in R with kernel J , such that f maps each A(k) onto B(k).The kernel J of the induced map gr(f) : gr(A) → gr(B) is a homogeneous ∂-ideal (i.e.,∂J ⊂ J). A set ai| i ∈ I such that ai is homogeneous of degree di is said to generate Jas a ∂-ideal if ∂kai| i ∈ I, k ≥ 0 generates J as an ideal.

Lemma 2.2. Let ai|i ∈ I be a generating set for J as a ∂-ideal, where ai is homogeneous ofdegree di. Then there exist elements ai(z) ∈ A(di) with ϕdi(ai(z)) = ai, such that ai(z)| i ∈ Igenerates J as a vertex algebra ideal.

We now define a good increasing filtration on the symplectic fermion algebra A(n).First, A(n) has a basis consisting of the normally ordered monomials

(2.6) : ∂I1e1 · · · ∂Inen∂J1f 1 · · · ∂Jnfn : .

In this notation, Ik = (ik1, . . . , ikrk) and Jk = (jk1 , . . . , j

ksk) are lists of integers satisfying

0 ≤ ik1 < · · · < ikrk and 0 ≤ jk1 < · · · < jksk , and

∂Ikek = : ∂ik1ek · · · ∂ikrkek :, ∂Jkfk = : ∂jk1 fk · · · ∂jkskfk : .

We have a Z≥0-grading

(2.7) A(n) =⊕d≥0

A(n)(d),

6

where A(n)(d) is spanned by monomials of the form (2.6) of total degree d =∑n

k=1 rk + sk.Finally, we define the filtration A(n)(d) =

⊕di=0A(n)(i). This filtration satisfies (2.4), and

we have an isomorphism of Sp(2n)-modules

(2.8) A(n) ∼= gr(A(n)),

and an isomorphism of graded supercommutative rings

(2.9) gr(A(n)) ∼=∧⊕

k≥0

Uk.

Here Uk is the copy of the standard Sp(2n)-module C2n with basis eik, f ik. In this nota-

tion, eik and f ik are the images of ∂kei(z) and ∂kf i(z) in gr(A(n)). The ∂-ring structure on∧⊕

k≥0 Uk is defined by ∂eik = eik+1 and ∂f ik = f i

k+1. For any reductive group G ⊂ Sp(2n),this filtration is G-invariant and is inherited by A(n)G. We obtain a linear isomorphismA(n)G ∼= gr(A(n)G) and isomorphisms of ∂-rings

(2.10) gr(A(n)G) ∼= gr(A(n))G ∼=(∧⊕

k≥0

Uk

)G.

The weight grading on A(n) is inherited by gr(A(n)) and (2.10) preserves weight as wellas degree, where wt(eik) = wt(f i

k) = k + 1.

3. THE STRUCTURE OF A(n)Sp(2n)

Recall Weyl’s first and second fundamental theorems of invariant theory for the stan-dard representation of Sp(2n) (Theorems 6.1.A and 6.1.B of [We]).

Theorem 3.1. For k ≥ 0, let Uk be the copy of the standard Sp(2n)-module C2n with symplecticbasis xi,k, yi,k| i = 1, . . . , n. Then (Sym

⊕k≥0 Uk)

Sp(2n) is generated by the quadratics

(3.1) qa,b =1

2

n∑i=1

(xi,ayi,b − xi,byi,a

), 0 ≤ a < b.

For a > b, define qa,b = −qb,a, and let Qa,b| a, b ≥ 0 be commuting indeterminates satisfyingQa,b = −Qb,a and no other algebraic relations. The kernel In of the homomorphism

(3.2) C[Qa,b] → (Sym⊕k≥0

Uk)Sp(2n), Qa,b 7→ qa,b,

is generated by the degree n + 1 Pfaffians pI , which are indexed by lists I = (i0, . . . , i2n+1) ofintegers satisfying

(3.3) 0 ≤ i0 < · · · < i2n+1.

For n = 1 and I = (i0, i1, i2, i3), we have

pI = qi0,i1qi2,i3 − qi0,i2qi1,i3 + qi0,i3qi1,i2 ,

and for n > 1 they are defined inductively by

(3.4) pI =2n+1∑r=1

(−1)r+1qi0,irpIr ,

where Ir = (i1, . . . , ir, . . . , i2n+1) is obtained from I by omitting i0 and ir.

7

There is an analogue of this theorem when the symmetric algebra Sym⊕

k≥0 Uk is re-placed by the exterior algebra

∧⊕k≥0 Uk. It is a special case of Sergeev’s first and second

fundamental theorems of invariant theory for Osp(m, 2n) (Theorem 1.3 of [SI] and Theo-rem 4.5 of [SII]). The generators of (

∧⊕k≥0 Uk)

Sp(2n) are

(3.5) qa,b =1

2

n∑i=1

(xi,ayi,b + xi,byi,a

), 0 ≤ a ≤ b.

For a > b, define qa,b = qb,a, and let Qa,b| a, b ≥ 0 be commuting indeterminates satisfy-ing Qa,b = Qb,a and no other algebraic relations. The kernel In of the homomorphism

(3.6) C[Qa,b] → (∧⊕

k≥0

Uk)Sp(2n), Qa,b 7→ qa,b,

is generated by elements pI of degree n + 1 which are indexed by lists I = (i0, . . . , i2n+1)satisfying

(3.7) 0 ≤ i0 ≤ · · · ≤ i2n+1.

For n = 1 and I = (i0, i1, i2, i3), we have

(3.8) pI = qi0,i1qi2,i3 + qi0,i2qi1,i3 + qi0,i3qi1,i2 ,

and for n > 1 they are defined inductively by

(3.9) pI =2n+1∑r=1

qi0,irpIr ,

where Ir = (i1, . . . , ir, . . . , i2n+1) is obtained from I by omitting i0 and ir.The generators qa,b of R correspond to vertex operators

(3.10) ωa,b =1

2

n∑i=1

(: ∂aei∂bf i : + : ∂bei∂af i :

), 0 ≤ a ≤ b,

of A(n)Sp(2n), satisfying ϕ2(ωa,b) = qa,b. By Lemma 2.1, ωa,b| 0 ≤ a ≤ b strongly generatesA(n)Sp(2n). In fact, there is a more economical strong generating set. For each m ≥ 0, letAm denote the vector space spanned by ωa,b| a + b = m, which has weight m + 2. Wehave dim(A2m) = m+ 1 = dim(A2m+1) for m ≥ 0, so

(3.11) dim(A2m/∂(A2m−1)

)= 1, dim

(A2m+1/∂(A2m)

)= 0.

For m ≥ 0, define

(3.12) j2m = ω0,2m,

which is clearly not a total derivative. We have

(3.13) A2m = ∂(A2m−1)⊕ ⟨j2m⟩ = ∂2(A2m−2)⊕ ⟨j2m⟩,where ⟨j2m⟩ is the linear span of j2m. Similarly,

(3.14) A2m+1 = ∂2(A2m−1)⊕ ⟨∂j2m⟩ = ∂3(A2m−2)⊕ ⟨∂j2m⟩.Moreover, ∂2ij2m−2i| 0 ≤ i ≤ m and ∂2i+1j2m−2i| 0 ≤ i ≤ m are bases of A2m andA2m+1, respectively, so each ωa,b ∈ A2m and ωc,d ∈ A2m+1 can be expressed uniquely as

(3.15) ωa,b =m∑i=0

λi∂2ij2m−2i, ωc,d =

m∑i=0

µi∂2i+1j2m−2i

8

for constants λi, µi. Hence j2m| m ≥ 0 is also a strong generating set for A(n)Sp(2n).

Theorem 3.2. A(n)Sp(2n) is generated by j0 and j2 as a vertex algebra.

Proof. It suffices to show that each j2k can be generated by these elements. This followsfrom the calculation

j2 1 j2k = −(2k + 4)j2k+2 + ∂2ω,

where ω is a linear combination of ∂2ij2k−2i for i = 0, . . . , k.

Consider the category of vertex algebras with generators J2m| m ≥ 0, which satisfythe same OPE relations as the generators j2m| m ≥ 0 of A(n)Sp(2n). Since the vectorspace with basis 1 ∪ ∂lj2m| l,m ≥ 0 is closed under n for all n ≥ 0, it forms a Lieconformal algebra. By Theorem 7.12 of [BK], this category contains a universal objectMn, which is freely generated by J2m|m ≥ 0. Then A(n)Sp(2n) is a quotient of Mn by anideal In, and since A(n)Sp(2n) is a simple vertex algebra, In is maximal. Let

πn : Mn → A(n)Sp(2n), J2m 7→ j2m

denote the quotient map. Using (3.15), which holds in A(n)Sp(2n) for all n, we can definean alternative strong generating set Ωa,b| 0 ≤ a ≤ b for Mn by the same formula: fora+ b = 2m and c+ d = 2m+ 1,

Ωa,b =m∑i=0

λi∂2iJ2m−2i, Ωc,d =

m∑i=0

µi∂2i+1J2m−2i.

Clearly πn(Ωa,b) = ωa,b. We shall use the same notation Am to denote the span of Ωa,b| a+b = m, when no confusion can arise. Note that Mn has a good increasing filtrationin which (Mn)(2k) is spanned by iterated Wick products of the generators J2m and theirderivatives, of length at most k, and (Mn)(2k+1) = (Mn)(2k). Equipped with this filtration,Mn lies in the category R, and πn is a morphism in R.

The structure of the ideal In. Under the identifications

gr(Mn) ∼= C[Qa,b], gr(A(n)Sp(2n)) ∼= (∧⊕

k≥0

Uk)Sp(2n) ∼= C[qa,b]/In,

gr(πn) is just the quotient map (3.2).

Lemma 3.3. For each I = (i0, i1, . . . , i2n+1), there exists a unique element

(3.16) PI ∈ (Mn)(2n+2) ∩ In

of weight 2n+ 2 +∑2n+1

a=0 ia, satisfying

(3.17) ϕ2n+2(PI) = pI .

These elements generate In as a vertex algebra ideal.

Proof. Clearly πn maps each (Mn)(k) onto (A(n)Sp(2n))(k), so the hypotheses of Lemma 2.2are satisfied. Since In = Ker(gr(πn)) is generated by pI, we can apply Lemma 2.2 tofind PI ∈ (Mn)(2n+2) ∩ In satisfying ϕ2n+2(PI) = pI , such that PI generates In. If P ′

I alsosatisfies (3.17), we would have PI − P ′

I ∈ (Mn)(2n) ∩ In. Since there are no relations inA(n)Sp(2n) of degree less than 2n+ 2, we have PI − P ′

I = 0. 9

Let ⟨PI⟩ denote the vector space with basis PI where I satisfies (3.3). We have

⟨PI⟩ = (Mn)(2n+2) ∩ In,

and clearly ⟨PI⟩ is a module over the Lie algebra P generated by J2m(k)| m, k ≥ 0. It isconvenient to work with a different generating set for P , namely

Ωa,b(a+ b+ 1− w)| 0 ≤ a ≤ b, a+ b+ 1− w ≥ 0.

Note that Ωa,b(a + b + 1 − w) is homogeneous of weight w, and P acts on gr(Mn) byderivations of degree zero. This action is independent of n, and is specified by the actionon the generators Ωc,d. We compute

(3.18) Ωa,b(a+ b+ 1− w)(Ωc,d) = λa,b,w,c(Ωc+w,d) + λa,b,w,d(Ωc,d+w),

where

(3.19) λa,b,w,c =

(−1)a+1 (a+c+1)!(c−b+w)!

+ (−1)b+1 (b+c+1)!(c−a+w)!

c− b+ w ≥ 0

0 c− b+ w < 0

.

The action of P on ⟨PI⟩ is by weighted derivation in the following sense. Given I =(i0, . . . , i2n+1) and p = Ωa,b(a+ b+ 1− w) ∈ P , we have

(3.20) p(PI) =2n+1∑r=0

λrPIr ,

where Ir = (i0, . . . , ir−1, ir + w, ir+1, . . . , i2n+1) and λr = λa,b,w,ir .For each n ≥ 1, there is a distinguished element P0 ∈ In, defined by

P0 = PI , I = (0, 0, . . . , 0).

It is the unique element of In of minimal weight 2n + 2, and is a singular vector in Mn.In fact, P0 generates In as a vertex algebra ideal. The proof is similar to the proof ofTheorem 7.2 of [LV], and involves using (3.18)-(3.20) to show that ⟨PI⟩ is generated by P0

as a module over P . This is sufficient because ⟨PI⟩ generates In as a vertex algebra ideal.

Normal ordering and quantum corrections. Given a homogeneous p ∈ gr(Mn) ∼= C[Qa,b]of degree k in the Qa,b, a normal ordering of p will be a choice of normally ordered polyno-mial P ∈ (Mn)(2k), obtained by replacing Qa,b by Ωa,b, and by replacing ordinary productswith iterated Wick products. For any choice we have ϕ2k(P ) = p. For the rest of thissection, P 2k, E2k, F 2k, etc., will denote elements of (Mn)(2k) which are homogeneous,normally ordered polynomials of degree k in the Ωa,b.

Let P 2n+2I ∈ (Mn)(2n+2) be some normal ordering of pI , so that ϕ2n+2(P

2n+2I ) = pI . Then

πn(P2n+2I ) ∈ (A(n)Sp(2n))(2n),

and ϕ2n(πn(P2n+2I )) ∈ gr(A(n)Sp(2n)) can be expressed uniquely as a polynomial of degree

n in the variables qa,b. Choose some normal ordering of the corresponding polynomial inthe variables Ωa,b, and call this element −P 2n

I . Then P 2n+2I + P 2n

I satisfies

ϕ2n+2(P2n+2I + P 2n

I ) = pI , πn(P2n+2I + P 2n

I ) ∈ (A(n)Sp(2n))(2n−2).

10

Continuing this process, we obtain an element∑n+1

k=1 P2kI in the kernel of πn, such that

ϕ2n+2(∑n+1

k=1 P2kI ) = pI . By Lemma 3.3, must have

(3.21) PI =n+1∑k=1

P 2kI .

The term P 2I lies in the space Am spanned by Ωa,b| a+ b = m, for m = 2n+

∑2n+1a=0 ia. By

(3.13), for all even integers m ≥ 1 we have a projection

prm : Am → ⟨Jm⟩.

For all I = (i0, i1, . . . , i2n+1) such that m = 2n+∑2n+1

a=0 ia is even, define the remainder

(3.22) RI = prm(P2I ).

Lemma 3.4. Fix PI ∈ In with I = (i0, i1, . . . , i2n+1) and m = 2n+∑2n+1

a=0 ia even. Suppose thatPI =

∑n+1k=1 P

2kI and PI =

∑n+1k=1 P

2kI are two different decompositions of PI of the form (3.21).

ThenP 2I − P 2

I ∈ ∂2(Am−2).

In particular, RI is independent of the choice of decomposition of PI .

Proof. The argument is the same as the proof of Lemma 4.7 and Corollary 4.8 of [LI].

Lemma 3.5. Let R0 denote the remainder of the element P0. The condition R0 = 0 is equivalentto the existence of a decoupling relation in A(n)Sp(2n) of the form

(3.23) j2n = Q(j0, j2, . . . , j2n−2),

where Q is a normally ordered polynomial in j0, j2, . . . , j2n−2 and their derivatives.

Proof. This is the same as the proof of Lemma 4.9 of [LI].

Lemma 3.6. Suppose that R0 = 0. Then for all m ≥ n, there exists a decoupling relation

(3.24) j2m = Qm(j0, j2, . . . , j2n−2).

Here Qm is a normally ordered polynomial in j0, j2, . . . , j2n−2, and their derivatives.

Proof. This is the same as the proof of Lemma 8.3 of [LV].

A recursive formula for RI . In this section we find a recursive formula for RI for anyI = (i0, i1, . . . , i2n+1) such that wt(PI) = 2n+2+

∑2n+1a=0 ia is even. It will be clear from our

formula that R0 = 0. We introduce the notation

(3.25) RI = Rn(I)Jm, m = 2n+

2n+1∑a=0

ia,

so that Rn(I) denotes the coefficient of Jm in prm(P2I ). For n = 1 and I = (i0, i1, i2, i3) the

following formula is easy to obtain using the fact that prm(Ωa,b) = (−1)mJm for m = a+ b.

11

R1(I) = −1

8

((−1)i0+i2 + (−1)i0+i3 + (−1)i1+i2 + (−1)i1+i3

2 + i0 + i1

+(−1)i0+i1 + (−1)i0+i3 + (−1)i1+i2 + (−1)i2+i3

2 + i0 + i2

+(−1)i0+i1 + (−1)i0+i2 + (−1)i1+i3 + (−1)i2+i3

2 + i1 + i2

+(−1)i0+i1 + (−1)i0+i2 + (−1)i1+i3 + (−1)i2+i3

2 + i0 + i3

+(−1)i0+i1 + (−1)i0+i3 + (−1)i1+i2 + (−1)i2+i3

2 + i1 + i3

+(−1)i0+i2 + (−1)i0+i3 + (−1)i1+i2 + (−1)i1+i3

2 + i2 + i3

).

(3.26)

Assume that Rn−1(J) has been defined for all J . Recall first that A(n) is a graded algebrawith Z≥0 grading (2.7), which specifies a linear isomorphism

A(n) ∼=∧⊕

k≥0

Uk, Uk∼= C2n.

Since A(n)Sp(2n) is a graded subalgebra of A(n), we obtain an isomorphism of gradedvector spaces

(3.27) in : A(n)Sp(2n) → (∧⊕

k≥0

Uk)Sp(2n).

Let p ∈ (∧⊕

k≥0 Uk)Sp(2n) be homogeneous of degree 2d, and let

f = (in)−1(p) ∈ (A(n)Sp(2n))(2d)

be the corresponding homogeneous element. Let F ∈ (Mn)(2d) be an element satisfyingπn(F ) = f , where πn : Mn → A(n)Sp(2n) is the projection. We can write F =

∑dk=1 F

2k,where F 2k is a normally ordered polynomial of degree k in the Ωa,b.

Next, consider the rank n+ 1 symplectic fermion algebra A(n+ 1), and let

qa,b ∈ (∧⊕

k≥0

Uk)Sp(2n+2) ∼= gr(A(n+ 1))Sp(2n+2) ∼= gr(A(n+ 1)Sp(2n+2))

be the generator given by (3.1), where Uk∼= C2n+2. Let p be the polynomial of degree 2d

obtained from p by replacing each qa,b with qa,b, and let

f = (in+1)−1(p) ∈ (A(n+ 1))Sp(2n+2))(2d)

be the corresponding homogeneous element. Finally, let F 2k ∈ Mn+1 be the elementobtained from F 2k by replacing each Ωa,b with the corresponding generator Ωa,b ∈ Mn+1,and let F =

∑di=1 F

2k.

12

Lemma 3.7. Fix n ≥ 1, and let PI be an element of In given by Lemma 3.3. There exists adecomposition PI =

∑n+1k=1 P

2kI of the form (3.21) such that the corresponding element

PI =n+1∑k=1

P 2kI ∈ Mn+1

has the property that πn+1(PI) lies in the homogeneous subspace (A(n+ 1)Sp(2n+2))(2n+2).

Proof. The argument is the same as the proof of Corollary 4.14 of [LI], and is omitted.

Recall that pI has an expansion pI =∑2n+1

r=1 qi0,irpIr , where Ir = (i1, . . . , ir, . . . , i2n+1) isobtained from I by omitting i0 and ir. Let PIr ∈ Mn−1 be the element corresponding topIr . By Lemma 3.7, there exists a decomposition

PIr =n∑

i=1

P 2iIr

such that the corresponding element PIr =∑n

i=1 P2iIr

∈ Mn has the property that πn(PIr)

lies in (A(n)Sp(2n))(2n). We have

(3.28)2n+1∑r=1

: Ωi0,ir PIr : =2n+1∑r=1

n∑i=1

: Ωi0,ir P2iIr : .

The right hand side of (3.28) consists of normally ordered monomials of degree at least2 in the generators Ωa,b, and hence contributes nothing to Rn(I). Since πn(PIr) is homo-geneous of degree 2n, πn(: Ωi0,ir PIr :) consists of a piece of degree 2n + 2 and a piece ofdegree 2n coming from all double contractions of Ωi0,ir with terms in PIr , which lower thedegree by two. The component of

πn

( 2n+1∑r=1

: Ωi0,ir PIr :

)∈ A(n)Sp(2n)

in degree 2n + 2 must cancel since this sum corresponds to pI , which is a relation amongthe variables qa,b. The component of : Ωi0,ir PIr : in degree 2n is

(3.29) Sr =1

2

((−1)i0+1

∑a

PIr,a

i0 + ia + 2+ (−1)ir+1

∑a

PIr,a

ir + ia + 2

)In this notation, for a ∈ i0, . . . , i2n+1\i0, ir, Ir,a is obtained from Ir = (i1, . . . , ir, . . . , i2n+1)by replacing ia with ia + i0 + ir + 2. It follows that

(3.30) πn

( 2n+1∑r=1

: Ωi0,ir PIr :

)= πn

( 2n+1∑r=1

Sr

).

Combining (3.28) and (3.30), we can regard

2n+1∑r=1

n∑i=1

: Ωi0,ir P2iIr : −

n∑r=0

Sr

13

as a decomposition of PI of the form PI =∑n+1

k=1 P2kI where the leading term P 2n+2

I =∑2n+1r=0 : Ωi0,ir P

2nIr

:. Therefore Rn(I) is the negative of the sum of the terms Rn−1(J)

corresponding to each PJ appearing in∑2n+1

r=0 Sr, so we obtain the following result.

Theorem 3.8. Rn(I) satisfies the recursive formula

(3.31) Rn(I) = −1

2

2n+1∑r=1

((−1)i0+1

∑a

Rn−1(Ir,a)

i0 + ia + 2+ (−1)ir+1

∑a

Rn−1(Ir,a)

ir + ia + 2

).

Now suppose that all the entries i0, . . . , i2n+1 appearing in I are even. Clearly eachIr,a appearing in (3.31) consists only of even entries as well. In the case n = 1 and I =(i0, i1, i2, i3), (3.26) reduces to

R1(I) = −1

2

(1

2 + i0 + i1+

1

2 + i0 + i2+

1

2 + i1 + i2+

1

2 + i0 + i3+

1

2 + i1 + i3+

1

2 + i2 + i3

),

so in particular R1(I) = 0. By induction on n, it is immediate from (3.31) that Rn(I) = 0whenever I has even entries.

Theorem 3.9. For all n ≥ 1, A(n)Sp(2n) has a minimal strong generating set j0, j2, . . . , j2n−2,and is therefore a W-algebra of type W(2, 4, . . . , 2n).

Proof. For I = (0, 0, . . . , 0), we have Rn(I) = 0, so R0 = Rn(I)J2n = 0. The claim then

follows from Lemma 3.6.

Using Theorem 3.9, it is now straightforward to describe A(mn)Sp(2n) for all m,n ≥ 1.Denote the generators of A(mn) by ei,j, f i,j for i = 1, . . . , n and j = 1, . . . ,m, which satisfy

ei,j(z)fk,l(w) ∼ δi,kδj,l(z − w)−2.

Under the action of Sp(2n), ei,j, f i,j| i = 1, . . . , n spans a copy of the standard Sp(2n)-module C2n for each j = 1, . . . ,m. Define

ωj,ka,b =

1

2

n∑i=1

(: ∂aei,j∂bf i,k : + : ∂bei,k∂af i,j :

).

By Theorem 3.1, ωj,ka,b| 1 ≤ j, k ≤ m, a, b ≥ 0 strongly generates A(mn)Sp(2n). In fact,

ωj,j0,2k| k ≥ 0, j = 1, . . . ,m

∪ωj,k

0,l 1 ≤ j < k ≤ m, l ≥ 0

also strongly generates A(mn)Sp(2n). This is clear because the corresponding elements ofgr(A(mn)Sp(2n)) generate gr(A(mn)Sp(2n)) as a ∂-ring.

Theorem 3.10. A(mn)Sp(2n) has a minimal strong generating set

(3.32) ωj,j0,2r| 1 ≤ j ≤ m, 0 ≤ r ≤ n− 1

∪ωj,k

0,s| 1 ≤ j < k ≤ m, 0 ≤ s ≤ 2n− 1.

Proof. By Theorem 3.9, we have decoupling relations

(3.33) ωj,j0,2r = Qr(ω

j,j0,0, . . . , ω

j,j0,2n−2),

for all r ≥ n and j = 1, . . . ,m. We now construct decoupling relations expressing eachωj,k0,a as a normally polynomial in the generators (3.32) and their derivatives, for j < k and

a ≥ 2n. We need the following calculations.

ωj,k0,0 1 ω

j,j0,2r = −(r + 1)ωj,k

0,2r + ∂ω, ωj,k0,0 1 ∂ω

j,j0,2r+1 = −ωj,k

0,2r+1 + ∂ν.

14

Here ω is a linear combination of ∂2r−tωj,k0,t for t = 0, . . . , 2r − 1, and ν is a linear combina-

tion of ∂2r−t+1ωj,k0,t for t = 0, . . . , 2r. Therefore applying the operator ωj,k

0,01 to (3.33) yieldsa relation

ωj,k0,2r = Qr(ω

j,j0,0, ω

j,j0,2, . . . ω

j,j0,2n−2, ω

j,k0,0, ω

j,k0,1, . . . , ω

j,k0,2n−1),

for all r ≥ n. Similarly, applying ωj,k0,01 to the derivative of (3.33) yields a relation

ωj,k0,2r+1 = Qr(ω

j,j0,0, ω

j,j0,2, . . . ω

j,j0,2n−2, ω

j,k0,0, ω

j,k0,1, . . . , ω

j,k0,2n−1),

for all r ≥ n. This shows that (3.32) is a strong generating set for A(mn)Sp(2n). It is minimalbecause there are no normally ordered relations of weight less than 2n+ 2.

4. THE STRUCTURE OF A(n)GL(n)

The subgroup GL(n) ⊂ Sp(2n) acts on A(n) such that the generators ei and f iof A(n) span copies of the standard GL(n)-modules Cn and (C∗)n, respectively. In thissection, we use a similar approach to find a minimal strong generating set for A(n)GL(n).First, we have isomorphisms

gr(A(n)GL(n)) ∼= gr(A(n))GL(n) ∼= R :=(∧⊕

j≥0

(Vj ⊕ V ∗j ))GL(n)

,

where Vj∼= Cn and V ∗

j∼= (Cn)∗ as GL(n)-modules. By an odd analogue of Weyl’s first

and second fundamental theorems of invariant theory for the standard representation ofGL(n), R is generated by the quadratics pa,b =

∑ni=1 x

iay

ib where xi

a is a basis for Va andyib is the dual basis for V ∗

b . The ideal of relations is generated by elements dI,J of degreen + 1, which are indexed by lists I = (i0, i1, . . . , in) and J = (j0, j1, . . . , jn) of integerssatisfying 0 ≤ i0 ≤ · · · ≤ in and 0 ≤ j0 ≤ · · · ≤ jn, which are analogous to determinantsbut without the signs. (This is a special case of Theorems 2.1 and 2.2 of [SII]). For n = 1,

dI,J = pi0,j0pi1,j1 + pi1,j0pi0,j1 ,

and for n > 1, dI,J is defined inductively by

dI,J =n∑

r=0

pir,j0dIr,J ′ ,

where Ir = (i0, . . . , ir . . . , in) is obtained from I by omitting ir, and J ′ = (j1, . . . , jn) isobtained from J by omitting j0.

The generators pa,b correspond to strong generators

γa,b =n∑

i=1

: ∂aei∂bf i :, a, b ≥ 0,

for A(n)GL(n), where wt(γa,b) = a + b + 2. Let Am be spanned by γa,b| a + b = m. Thendim(Am) = m+ 1 and dim(Am/∂Am−1) = 1, so

Am = ∂Am−1 ⊕ ⟨hm⟩, hm = γ0,m.

Since ∂ahm−a| a = 0, . . . ,m and γa,m−a| a = 0, . . . ,m are both bases for Am, hk| k ≥ 0strongly generates A(n)GL(n).

Lemma 4.1. A(n)GL(n) is generated as a vertex algebra by h0 and h1.

15

Proof. This is immediate from the following calculation.

h1 1 hk = −(k + 3)hk+1 + 2∂hk, k ≥ 1.

There exists a vertex algebra Nn which is freely generated by Hk| k ≥ 0 with sameOPE relations as hk| k ≥ 0, such that A(n)GL(n) is a quotient of Nn by an ideal Jn undera map

πn : Nn → A(n)GL(n), Hk → hk.

We have an alternative strong generating set Γa,b| a, b ≥ 0 for Nn satisfying πn(Γa,b) =γa,b. There is a good increasing filtration on Nn where (Nn)(2k) is spanned by iterated Wickproducts of Hm and their derivatives of length at most k, and (Nn)(2k+1) = (Nn)(2k), andπn preserves filtrations.

Lemma 4.2. For each I and J as above, there exists a unique element DI,J ∈ (Nn)(2n+2) ∩ Jn ofweight 2n + 2 +

∑n+1a=0(ia + ja), satisfying ϕ2n+2(DI,J) = dI,J . These elements generate Jn as a

vertex algebra ideal.

Each DI,J can be written in the form

(4.1) DI,J =n+1∑k=1

D2kI,J ,

where D2kI,J is a normally ordered polynomial of degree k in the generators Γa,b. The term

D2I,J lies in the space Am for m = 2n +

∑na=0(ia + ja). We have the projection prm : Am →

⟨Hm⟩, and we define the remainder

RI,J = prm(D2I,J).

It is independent of the choice of decomposition (4.1).The element of Jn of minimal weight corresponds to I = (0, . . . , 0) = J , and has weight

2n + 2. We denote this element by D0 and we denote its remainder by R0. The conditionR0 = 0 is equivalent to the existence of a decoupling relation

(4.2) h2n = P (h0, h1, . . . , h2n−1),

where P is a normally ordered polynomial in h0, h1, . . . , h2n−1 and their derivatives. Fromthis relation it is easy to construct decoupling relations hm = Pm(h

0, h1, . . . , h2n−1) forall m > 2n, so the condition R0 = 0 implies that h0, h1, . . . , h2n−1 is a minimal stronggenerating set for A(n)GL(n).

To prove this, we need to analyze the quantum corrections of DI,J . Write

RI,J = Rn(I, J)Hm, m = 2n+

n∑a=0

(ia + ja),

so that Rn(I, J) denotes the coefficient of Hm in prm(D2I,J). For n = 1 and I = (i0, i1),

J = (j0, j1) we have

(4.3) R1(I, J) = (−1)1+j0+j1

(1

2 + i0 + j0+

1

2 + i1 + j0+

1

2 + i0 + j1+

1

2 + i1 + j1

).

Using the same method as the previous section, one can show that Rn(I, J) satisfies thefollowing recursive formula.

16

(4.4) Rn(I, J) =n∑

r=0

((−1)j0

(∑k

Rn−1(Ir,k, J′)

2 + ik + j0

)+ (−1)ir

∑l

(Rn−1(Ir, J

′l )

2 + jl + ir

)).

In this notation, Ir = (i0, . . . , ir, . . . , in) is obtained from I by omitting ir. For k = 0, . . . , nand k = r, Ir,k is obtained from Ir by replacing the entry ik with ik + ir + j0 + 2. Similarly,J ′ = (j1, . . . , jn) is obtained from J by omitting j0, and for l = 1, . . . , n, J ′

l is obtained fromJ ′ by replacing jl with jl + ir + j0 + 2.

Suppose that all entries of I and J are even. Then for each Rn−1(K,L) appearing in(4.4), all entries of K and L are even. It is immediate from (4.3) that R1(I, J) = 0, and byinduction on n, it follows from (4.4) that Rn(I, J) is nonzero whenever I and J consist ofeven numbers. Specializing to the case I = (0, . . . , 0) = J , we see that R0 = 0, as desired.Finally, this implies

Theorem 4.3. A(n)GL(n) has a minimal strong generating set h0, h1, . . . , h2n−1, and is there-fore of type W(2, 3, . . . , 2n+ 1).

An immediate consequence is

Theorem 4.4. For all m ≥ 1, A(mn)GL(n) has a minimal strong generating set

γj,k0,l =

n∑i=1

: ei,j∂lf i,k :, 1 ≤ j ≤ k ≤ m, 0 ≤ l ≤ 2n− 1.

Proof. The argument is similar to the proof of Theorem 3.10, and is omitted.

5. CHARACTER DECOMPOSITIONS

A general result of Kac and Radul (see Section 1 of [KR]) states that for an associa-tive algebra A and a Lie algebra g, every (g, A)-module V with the properties that it isan irreducible A-module and a direct sum of a countable number of finite-dimensionalirreducible g-modules decomposes as

V ∼=⊕E

(E ⊗ V E

),

where the sum is over all irreducible g-modules E and V E is an irreducible Ag-module.Kac and Radul then remark that the same statement is true if we replace g by a group G(see also [DLMI] for similar results). The symplectic fermion algebra A(mn) is simple andgraded by conformal weight. Each graded subspace is finite-dimensional and G-invariantfor any reductive G ⊂ Sp(2mn). Hence the assumptions of [KR] apply, so that

A(mn) ∼=⊕E

(E ⊗A(mn)E

),

where the sum is over all finite-dimensional irreducible representations of G, and A(mn)E

is an irreducible representation of A(mn)G. The purpose of this section is to find thecharacters of the A(mn)E for G = Sp(2n) × SO(n) and also for G = GL(m) × GL(n).We will use the denominator identities of finite-dimensional classical Lie superalgebrasto solve this problem. We need some notation and results on Lie superalgebras. For this,we use the book [CW], and the article [KWII].

17

Define the lattice

Lm,n := δ1Z⊕ · · · ⊕ δmZ⊕ ϵ1Z⊕ · · · ⊕ ϵnZof signature (m,n), where the bilinear product is defined by

(δi, δj) = δi,j, (ϵa, ϵb) = −δa,b, (δi, ϵb) = 0,

for all 1 ≤ i, j ≤ m and 1 ≤ a, b ≤ n. We restrict to the case m > n. The root systemsof various Lie superalgebras can be constructed as subsets of Lm,n. We are interested inthree examples.

Example 5.1. Let g = gl (m|n). Then the root system ∆ is the disjoint union of even andodd roots, ∆ = ∆0 ∪∆1, where

∆0 = δi − δj, ϵa − ϵb | 1 ≤ i, j ≤ m, 1 ≤ a, b ≤ n ∆1 = ± (δi − ϵa) | 1 ≤ i ≤ m, 1 ≤ a ≤ n .

A Lie superalgebra usually allows for inequivalent choices of positive roots and positivesimple roots. We are interested in a choice of simple positive roots that has as many oddisotropic roots as possible. This is if m ≤ n

Π = (ϵa − δa) , δb − ϵb+1 |1 ≤ a, b ≤ m ∪ ϵi − ϵi+1 |m+ 1 ≤ i ≤ n so that positive roots are

∆+ = ∆+0 ∪∆+

1

∆+0 = δi − δj, ϵa − ϵb | 1 ≤ i < j ≤ m, 1 ≤ a < b ≤ n

∆+1 = ϵa − δi, δj − ϵb | 1 ≤ a ≤ n, a ≤ i ≤ m, 1 ≤ j ≤ m, j < b ≤ n

and

(5.1) S = (ϵa − δa) | 1 ≤ a ≤ m ⊂ Π

is a maximal isotropic subset of simple positive odd roots. The case m > n is analogous.For every positive root, define the Weyl reflection

rα(x) := x− 2(x, α)

(α, α)α.

The Weyl group W is then the group generated by the rα for α in ∆+0 . Finally, let W ♯ be

the subgroup of the Weyl group generated by the roots of gl(m) if m > n, and otherwisethe subgroup generated by the roots of gl(n).

Example 5.2. Let g = spo (2m|2n+ 1). Then the odd and even roots of the root system∆ = ∆0 ∪∆1 are

∆0 = ±δi ± δj, ±ϵa ± ϵb, ±2δp, ±ϵq , ∆1 = ±δp ± ϵq, ±δp ,

where 1 ≤ i < j ≤ m, 1 ≤ a < b ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n. A choice of simple positiveroots that has as many odd isotropic roots as possible is

Π = (−1)a (ϵa − δa) , ϵb − ϵb+1, δi − δi+1, δm |1 ≤ a, b ≤ m, b odd, 1 ≤ i ≤ n, i evenso that positive roots ∆+ = ∆+

0 ∪∆+1 split into even and odd roots as

∆+0 = δi ± δj, ϵa ± ϵb, 2δp, ϵq

∆+1 = ϵa ± δi, δj ± ϵb, δp | 1 ≤ a ≤ n, a ≤ i ≤ m, a even, 1 ≤ j ≤ m, j ≤ b ≤ n, j odd

18

where 1 ≤ i < j ≤ m, 1 ≤ a < b ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n unless otherwise indicated.Then (5.1) is a maximal isotropic subset of simple positive odd roots. The Weyl group Wis again the Weyl group of the even subalgebra. If m > n, W ♯ is defined to be the Weylgroup of sp(2m), and otherwise it is the Weyl group of so(2n+ 1).

Example 5.3. Let g = spo (2m|2n). Then the root system ∆ is the disjoint union of

∆0 = ±δi ± δj, ±ϵa ± ϵb, ±2δp , ∆1 = ±δp ± ϵq ,

where 1 ≤ i < j ≤ m, 1 ≤ a < b ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n. In this case, a choice ofsimple positive roots that has as many odd isotropic roots as possible is

Π = (−1)a (ϵa − δa) , ϵb − ϵb+1, δi − δi+1 |1 ≤ a, b ≤ m, b odd, 1 ≤ i ≤ n, i evenso that positive roots are the disjoint union of

∆+0 = δi ± δj, ϵa ± ϵb, 2δp

∆+1 = ϵa ± δi, δj ± ϵb | 1 ≤ a ≤ n, a ≤ i ≤ m, a even, 1 ≤ j ≤ m, j ≤ b ≤ n, j odd

where 1 ≤ i < j ≤ m, 1 ≤ a < b ≤ n, 1 ≤ p ≤ m, 1 ≤ q ≤ n unless otherwise indicated.Then again (5.1) is a maximal isotropic subset of simple positive odd roots. The Weylgroup W is as before the Weyl group of the even subalgebra. For m > n, W ♯ is the Weylgroup of sp(2m), and otherwise it is the Weyl group of so(2n).

The Weyl vector of a Lie superalgebra is defined to be ρ := ρ0 − ρ1 with even and oddWeyl vectors

ρ0 =1

2

∑α∈∆+

0

α, ρ1 =1

2

∑α∈∆+

1

α.

We denote the root lattice of g by L. Note that this lattice is spanned by the odd roots inour examples. In order to state the main result of this section, we need to define for everyλ in L the set

Iλ := (

(nα)α∈∆+1 \S , (mβ)β∈S

)∈ Z|∆+

1 |∣∣∣ ∑

α∈∆+1 \S

∑β∈S

nαα +mββ = λ

as well as the partial theta function

Pn(q) := q12(n+

12)

2∞∑

m=0

(−1)mq12(m2−2m(n+ 1

2)).

Theorem 5.4. Let g0 = gl(m) ⊕ gl(r) or g0 = sp(2m) ⊕ so(r) be the even subalgebra of g =gl(m|r), respectively g = spo(2m|r). Let P+ be the set of dominant weights of the even subalgebrag0. Then the graded character of A(mr) decomposes as

ch[A(mr)] =∑

Λ∈L∩P+

chΛBΛ.

Here chΛ is the irreducible highest-weight representation of g0 of highest-weight Λ. The branchingfunction is

BΛ =1

η(q)|∆+1 |

|W ♯||W |

∑ω∈W

ϵ(w)∑

((nα),(mβ))∈Iω(Λ+ρ0)−ρ0

q12(nα+

12)

2

Pmβ(q)

19

with the Dedekind eta function η(q) = q124

∞∏n=1

(1− qn).

Proof. Observe that the generators ei, f i|i = 1, . . . ,mr of the symplectic fermion algebraA(mr) carry the same representation of g0 as the odd subalgebra of g. Hence we can writethe graded character of A(mr) as

ch[A(mr)] = qmr12

∏α∈∆1

∞∏n=1

(1 + eαqn) ,

where ∆1 is the odd root system of g ∈ spo(2m|r), gl(m|r). Using the odd Weyl vector,the character can be rewritten as

ch[A(mr)] = e−ρ1qmr12

∏α∈∆+

1

eα2

(1 + e−α)

∞∏n=1

(1 + eαqn)(1 + e−αqn−1

)=

1

eρ1

∏α∈∆+

1

ϑ (eα; q)

(1 + e−α) η(q),

where η(q) is the Dedekind eta function and

ϑ(z; q) =∑n∈Z

zn+12 q

12(n+

12)

2

is a standard Jacobi theta function. The denominator identity of g is [G, KWII],

(5.2)

eρ0∏

α∈∆+0

(1− e−α)

eρ1∏

α∈∆+1

(1 + e−α)=∑ω∈W ♯

ϵ(ω)ω

eρ∏β∈S

(1 + e−β)

where W ♯ is the Weyl group of the larger even subalgebra of g as defined in Examples5.1–5.3. Here ϵ(ω) denotes the signum of the Weyl reflection ω. The set S is the set ofsimple isotropic roots introduced in the examples. The left-hand side of (5.2) is invariantunder ϵ(ω)ω for any reflection ω ∈ W . Hence we can rewrite the denominator identity as

eρ0∏

α∈∆+0

(1− e−α)

eρ1∏

α∈∆+1

(1 + e−α)=

|W ♯||W |

∑ω∈W

ϵ(ω)ω

eρ∏β∈S

(1 + e−β)

.

Observe that∏

α∈∆+1

ϑ (eα; q) is invariant under W , so we can rewrite the character as

ch[A(mr)] =1

η(q)|∆+1 |

|W ♯||W |

∑α∈∆+

1

∑nα∈Z

∑ω∈W

ϵ(ω)ω

eρ0enααq12(nα+

12)

2∏β∈S

(1 + e−β)

eρ0

∏α∈∆+

0

(1− e−α)

−1

.

For λ in L, we define the set

Jλ :=

((nα)α∈∆+1, (mβ)β∈S

)∈ Z|∆+

1 | × Z|S|≥0

∣∣∣ ∑α∈∆+

1

∑β∈S

nαα+mββ = λ

20

and we also define

chλ :=

∑ω∈W

ϵ(ω)ω(eλ+ρ0

)eρ0

∏α∈∆+

0

(1− e−α).

Up to a sign this is the character chΛ of the highest-weight representation of highest-weight Λ, where Λ is the dominant weight in the Weyl orbit of λ. Using this notation, thegraded character becomes

ch[A(mr)] =1

η(q)|∆+1 |

|W ♯||W |

∑λ∈L

chλ

∑((nα),(mβ))∈Jλ

(−1)mβq12(nα+

12)

2

=1

η(q)|∆+1 |

∑Λ∈L∩P+

chΛ

∑ω∈W

|W ♯||W |

ϵ(w)∑

((nα),(mβ))∈Jω(Λ+ρ0)−ρ0

(−1)mβq12(nα+

12)

2

where P+ is the set of dominant weights of the even subalgebra of g. If α = β ∈ S, thenwe can combine contributions as follows. Let ℓβ = nβ +mβ , then

∞∑mβ=0

(−1)mβq12(nβ+

12)

2

=∞∑

mβ=0

(−1)mβq12(ℓβ−mβ+

12)

2

= q12(ℓβ+

12)

2∞∑

mβ=0

(−1)mβq12(m2

β−2mβ(ℓβ+ 12))

Inserting this terminates the proof. Corollary 5.5. For G = Sp(2n), the branching function is

BΛ(q) =1

η(q)n

∑ω∈W

ϵ(ω)q12(ω(Λ+ρ0)−ρ,ω(Λ+ρ0)−ρ).

Proof. We have |∆+1 | = n and W ♯ = W . The set S is empty and the set IΛ (Λ in L) is just

the one element set Λ. For Λ = m1δ1 + · · ·+mnδn, we haven∑

i=1

(mi +

1

2

)2

= (Λ + ρ1,Λ + ρ1) ,

so that the branching function simplifies as claimed.

Corollary 5.6. A(n)Sp(2n) has graded character

ch[A(n)Sp(2n)] = qn12

∑m≥0

dim(A(n)Sp(2n)[m])qm = qn12

n∏i=1

∏k≥0

1

1− q2i+k.

Proof. Using the denominator identity of sp(2n), we get

B0 =1

η(q)n

∑ω∈W

ϵ(ω)q12(ω(ρ0)−ρ,ω(ρ0)−ρ) =

q12((ρ0,ρ0)+(ρ,ρ))

η(q)n

∑ω∈W

ϵ(ω)q−(ω(ρ0),ρ)

=q

12(ρ1,ρ1)

η(q)n

∏α∈∆+

0

(1− q(α,ρ)

).

The claim follows from the products of root vectors

(ρ, δi + δj) = 2n+ 1− i− j, (ρ, δi − δj) = j − i, (ρ, 2δi) = 2n+ 1− 2i,

21

and the norm (ρ1, ρ1) =n4

of the odd Weyl vector.

Corollary 5.7. A(n)Sp(2n) is freely generated by j0, j2, . . . , j2n−2. In other words, there are nonormally ordered polynomial relations among these generators and their derivatives.

We turn to the case of G = GL(m). Note that A(mn)GL(n)×GL(m) = A(mn)GL(n)×SL(m)

since one of the GL(1) factors acts trivially. For the case gl(m|1), we take as positive oddroots the set ϵ− δ1, δ2 − ϵ, δ3 − ϵ, . . . , δm − ϵ and we take S = ϵ− δ1. The set Iλ has onlyone element, and it is a short computation to obtain the following explicit expression forthe branching functions.

Corollary 5.8. Let G = GL(m) and Λ = Λ1δ1 + · · ·+ Λmδm. Then the branching function is

BΛ(q) =q−

(m−2)2

4

η(q)m

∑ω∈W

ϵ(ω)q12(ω(Λ+ρ0)−ρ,ω(Λ+ρ0)−ρ)

∞∑r=0

(−1)rqr(r−1)

2 q(δ1,ω(Λ+ρ0)−ρ0)(r+1).

Remark 5.9. Asymptotic properties of partial theta functions, like the one appearing in the char-acter decomposition, have been studied recently in [BM, CMW]. Especially in both works withdifferent methods it has been found that

limq→1

Pn(q) =1

2, for all n ∈ Z.

We also remark that these partial theta functions are not mock modular forms, but they are quan-tum modular forms as defined by Zagier [Za].

Corollary 5.10. A(n)GL(n) is not freely generated by h0, h1, . . . , h2n−1.

Proof. By the previous remark and corollary we get

limq→1

∣∣∣η(q)nB0(q)∣∣∣ ≤ ∑

ω∈W

1

2=

n!

2.

On the other hand, if A(n)GL(n) is freely generated its character would be2n+1∏i=2

∏k≥0

(1− qi+k)−1.

But the latter has a different behavior as q approaches one, namely

limq→1

(η(q)n

2n+1∏i=2

∏k≥0

(1− qi+k)−1

)= ∞,

so that the two cannot coincide.

6. THE HILBERT PROBLEM FOR A(n)

For a simple, strongly finitely generated vertex algebra V and a reductive group G ⊂Aut(V), the Hilbert problem asks whether VG is strongly finitely generated. In this sectionwe solve this problem affirmatively for V = A(n). The approach is similar to the one usedfor the βγ-system and free fermion algebra in [LV], and some details are omitted.

22

Recall from [Zh] that the Zhu functor assigns to a vertex algebra V =⊕

n∈Z Vn an asso-ciative algebra A(V) and a surjective linear map

πZhu : V → A(V), a 7→ [a].

A Z≥0-graded module M =⊕

n≥0 Mn over W is called admissible if for every a ∈ Vm,a(n)Mk ⊂ Mm+k−n−1, for all n ∈ Z. Given a ∈ Vm, a(m − 1) acts on each Mk. ThenM0 is an A(V)-module with action [a] 7→ a(m − 1) ∈ End(M0), and M 7→ M0 providesa one-to-one correspondence between irreducible, admissible V-modules and irreducibleA(V)-modules. If A(V) is commutative, all its irreducible modules are one-dimensional.The corresponding V-modules M are therefore cyclic and will be called highest-weightmodules.

Lemma 6.1. For n ≥ 1, A(A(n)Sp(2n)) is commutative.

Proof. Since A(n)Sp(2n) is strongly generated by j0, j2, . . . , j2n−2, A(A(n)Sp(2n)) is generatedby a0, a2, . . . , a2n−2 where a2m = πZhu(j

2m). The commutativity of A(A(n)Sp(2n)) for n = 1is clear since there is only one generator, and it is also clear for n = 2 since a0 is central,and hence commutes with a2. It follows that for n = 2, [a2i, a2j] = 0 where i, j ≥ 0. Sincethe nonconstant terms appearing in the operator product a2i(z)a2j(w) are independent ofn, it follows that [a2i, a2j] = 0 for all n.

Corollary 6.2. For n ≥ 1, A(A(n)Sp(2n)) ∼= C[a0, a2, . . . , a2n−2], so the irreducible, admissibleA(n)Sp(2n)-modules are all highest-weight modules, and are parametrized by the points in Cn.

Proof. This is immediate from Lemma 6.1 and Corollary 5.7.

Remark 6.3. A(A(n)GL(n)) is also commutative, but we shall not need this fact.

Recall that A(n) ∼= gr(A(n)) as Sp(2n)-modules, and

gr(A(n)G) ∼= (gr(A(n))G ∼= (∧⊕

k≥0

Uk)G = R

as supercommutative algebras, where Uk∼= C2n. For all p ≥ 1, GL(p) acts on

⊕p−1k=0 Uk and

commutes with the action of G. The inclusions GL(p) → GL(p + 1) induce an action ofGL(∞) = limp→∞ GL(p) on

⊕k≥0 Uk, so GL(∞) acts on

∧⊕k≥0 Uk and commutes with

the action of G. Therefore GL(∞) acts on R as well. Elements σ ∈ GL(∞) are known aspolarization operators, and σf is known as a polarization of f ∈ R.

Theorem 6.4. There exists an integer m ≥ 0 such that R is generated by the polarizations of anyset of generators for (

∧⊕mk=0 Uk)

G. Since G is reductive, (∧⊕m

k=0 Uk)G is finitely generated, so

there exists a finite set f1, . . . , fr, whose polarizations generate R.

Proof. It suffices to show that the degrees of the generators of (∧⊕

k≥0 Uk)G have an upper

bound d, since we can then take m = d. By Theorem 2.5A of [We], (Sym⊕

k≥0 Uk)G is

generated by the polarizations of any set of generators of (Sym⊕2n−1

k=0 Uk)G. Since G is

reductive (Sym⊕2n−1

k=0 Uk)G is finitely generated, and since polarization preserves degree,

the degrees of the generators of (Sym⊕

k≥0 Uk)G have an upper bound d.

23

For any vector space W , recall that∧r W is the quotient of the rth tensor power T rW

under the antisymmetrization map. We may regard T rW as the homogeneous subspace

Sym1(W1)⊗ · · · ⊗ Sym1(Wr) ⊂ Symr⊕

j=1

Wj, Wj∼= W

of degree 1 in each Wj . Suppose that G acts on W , and that the generators of Sym⊕

j≥1 Wj

have upper bound d. The symmetric group Sr acts on T rW by permuting the factors, and(T rW )G is generated by the elements of (T jW )G for j ≤ d, together with their translatesunder Sr. By taking W =

⊕k≥0 Uk and applying the antisymmetrization map, it follows

that (∧⊕

k≥0 Uk)G is generated by the elements of degree at most d.

Corollary 6.5. A(n)G has a strong generating set which lies in the direct sum of finitely manyirreducible A(n)Sp(2n)-submodules of A(n).

Proof. This is a straightforward consequence of Corollary 6.2, Theorem 6.4, and the fact(see Remark 1.1 of [KR]) that A(n) has a Howe pair decomposition

(6.1) A(n) ∼=⊕ν∈H

L(ν)⊗M ν ,

where H indexes the irreducible, finite-dimensional representations L(ν) of Sp(2n), andthe M ν are inequivalent, irreducible highest-weight A(n)Sp(2n)-modules. The argument issimilar to the proof of Lemma 2 of [LII] and is omitted.

Using the strong finite generation of A(n)Sp(2n) together with some finiteness propertiesof the irreducible, highest-weight A(n)Sp(2n)-modules appearing in A(n), we obtain thefollowing result. This is analogous to Lemma 9 of [LII], and the proof is omitted.

Lemma 6.6. Let M be an irreducible, highest-weight A(n)Sp(2n)-submodule of A(n). Given asubset S ⊂ M, let MS ⊂ M denote the subspace spanned by elements of the form

: ω1(z) · · ·ωt(z)α(z) :, ωj(z) ∈ A(n)Sp(2n), α(z) ∈ S.

There exists a finite set S ⊂ M such that M = MS .

Theorem 6.7. For any reductive group G of automorphisms of A(n), A(n)G is strongly finitelygenerated.

Proof. By Corollary 6.5, we can find f1(z), . . . , fr(z) ∈ A(n)G such that gr(A(n))G is gener-ated by the corresponding polynomials f1, . . . , fr ∈ gr(A(n))G, together with their polar-izations. We may assume that each fi(z) lies in an irreducible, highest-weight A(n)Sp(2n)-module Mi of the form L(ν)µ0⊗Mν , where L(ν)µ0 is a trivial, one-dimensional G-module.Furthermore, we may assume that f1(z), . . . , fr(z) are highest-weight vectors for the ac-tion of A(n)Sp(2n). For each Mi, choose a finite set Si ⊂ Mi such that Mi = (Mi)Si

, usingLemma 6.6. Define

S = j0, j2, . . . , j2n−2 ∪( r∪i=1

Si

).

Since j0, j2, . . . , j2n−2 strongly generates A(n)Sp(2n), and⊕r

i=1Mi contains a strong gen-erating set for A(n)G, it follows that S is a strong, finite generating set for A(n)G.

24

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DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ALBERTA

E-mail address: creutzig@ualberta.ca

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF DENVER

E-mail address: andrew.linshaw@du.edu

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